Question: All of the 5th grade teachers and students from Santa Rita went on a field trip to an art museum. Tickets were $$8.00$ each for teachers and $$4.50$ each for students, and the group paid $$64.50$ in total. The next month, the same group visited a science museum where the tickets cost $$16.00$ each for teachers and $$10.50$ each for students, and the group paid $$142.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+4.5y = 64.5}$ ${16x+10.5y = 142.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-16x-9y = -129}$ ${16x+10.5y = 142.5}$ Add the top and bottom equations together. $ 1.5y = 13.5 $ $ y = \dfrac{13.5}{1.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {8x+4.5y = 64.5}$ to find $x$ ${8x + 4.5}{(9)}{= 64.5}$ $8x+40.5 = 64.5$ $8x = 24$ $x = \dfrac{24}{8}$ ${x = 3}$ You can also plug ${y = 9}$ into $ {16x+10.5y = 142.5}$ and get the same answer for $x$ ${16x + 10.5}{(9)}{= 142.5}$ ${x = 3}$ There were $3$ teachers and $9$ students on the field trips.